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Overview of part 1 Historical perspective –From reductionism to systems and networks Examples of complex networks Structural/topological metrics –Average path length –Degree distribution –Clustering Topological models –Regular, random, small-world, scale-free networks –An evolutionary model of network growth: Preferential attachment Implications of scale-free property in: –Robustness/fragility –Epidemics/diffusion processes Focusing on the “small scale”: network motifs –Networks as functioning circuits

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References I used many slides from other talks: –S.Maslov, “Statistical physics of complex networks” –I.Yanai, “Evolution of networks” –D.Bonchev, “Networks basics” –Eileen Kraemer, “Topology and dynamics of complex networks”

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Historical perspective In the beginning.. there was REDUCTIONISM –All we need to know is the behavior of the system elements –Particles in physics, molecules or proteins in biology, communication links in the Internet –Complex systems are nothing but the result of many interactions between the system’s elements –No new phenomena will emerge when we consider the entire system –A centuries-old very flawed scientific tradition..

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Historical perspective During the 80’s and early 90’s, several parallel approaches departed from reductionism Consider the entire SYSTEM attempting to understand/explain its COMPLEXITY –B. Mandelbrot and others: Chaos and non-linear dynamical systems (the math of complexity) –P. Bak: Self-Organized Criticality – The edge of chaos –S. Wolfram: Cellular Automata –S. Kauffman: Random Boolean Networks –I. Prigogine: Dissipative Structures –J. Holland: Emergence –H. Maturana, F. Varela: Autopoiesis networks & cognition –Systems Biology

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Historical perspective Systems approach: thinking about Networks –The focus moves from the elements (network nodes) to their interactions (network links) –To a certain degree, the structural details of each element become less important than the network of interactions –Some system properties, such as Robustness, Fragility, Modularity, Hierarchy, Evolvability, Redundancy (and others) can be better understood through the Networks approach Some milestones: –1998: Small-World Networks (D.Watts and S.Strogatz) –1999: Scale-Free Networks (R.Albert & A.L.Barabasi) –2002: Network Motifs (U.Alon)

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Things derive their being and nature by mutual dependence and are nothing in themselves. -Nagarjuna, second century Buddhist philosopher An elementary particle is not an independently existing, unanalyzable entity. It is, in essence, a set of relationships that reach outward to other things. -H.P. Stapp, twentieth century physicist Some relevant Zen: See slides by Itay Yanai

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Overview of part 1 Historical perspective –From reductionism to systems and networks Examples of complex networks Structural/topological metrics –Average path length –Degree distribution –Clustering Topological models –Regular, random, small-world, scale-free networks –An evolutionary model of network growth: Preferential attachment Implications of scale-free property in: –Robustness/fragility –Epidemics/diffusion processes Focusing on the “small scale”: network motifs –Networks as functioning circuits

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Air Transportation Network

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Actors’ web Kraemer

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Mathematicians & Computer Scientists Kraemer

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Sexual contacts: M. E. J. Newman, The structure and function of complex networks, SIAM Review 45, (2003).

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High school dating: Data drawn from Peter S. Bearman, James Moody, and Katherine Stovel visualized by Mark Newman

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Internet as measured by Hal Burch and Bill Cheswick's Internet Mapping Project.Internet Mapping Project

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KEGG database: Metabolic networks

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Transcription regulatory networks Bacterium: E. coli Single-celled eukaryote: S. cerevisiae

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protein-gene interactions protein-protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio-chemical reactions Bio-Map L-A Barabasi miRNA regulation? _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

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C. elegans neuronal net

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Freshwater food web by Neo Martinez and Richard Williams

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Overview of part 1 Historical perspective –From reductionism to systems and networks Examples of complex networks Structural/topological metrics –Average path length –Degree distribution –Clustering Topological models –Regular, random, small-world, scale-free networks –An evolutionary model of network growth: Preferential attachment Implications of scale-free property in: –Robustness/fragility –Epidemics/diffusion processes Focusing on the “small scale”: network motifs –Networks as functioning circuits

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Networks As Graphs Networks can be undirected or directed, depending on whether the interaction between two neighboring nodes proceeds in both directions or in only one of them, respectively. The specificity of network nodes and links can be quantitatively characterized by weights Vertex-WeightedEdge-Weighted Bonchev

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Networks As Graphs - 2 Networks having no cycles are termed trees. The more cycles the network has, the more complex it is. A network can be connected (presented by a single component) or disconnected (presented by several disjoint components). connecteddisconnected trees cyclic graphs Bonchev

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Networks As Graphs - 3 Some Basic Types of Graphs Paths Stars Cycles Complete Graphs Bipartite Graphs Bonchev

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Structural metrics: Average path length Slides by Kraemer & Barabasi, Bonabeau (SciAm’03)

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Structural Metrics: Degree distribution(connectivity)

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Structural Metrics: Clustering coefficient

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Several other graph metrics exist We will study them as needed –Centrality –Betweenness –Assortativity –Modularity –…

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Network Evolution Slide by Kraemer

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Overview of part 1 Historical perspective –From reductionism to systems and networks Examples of complex networks Structural/topological metrics –Average path length –Degree distribution –Clustering Topological models –Regular, random, small-world, scale-free networks –An evolutionary model of network growth: Preferential attachment Implications of scale-free property in: –Robustness/fragility –Epidemics/diffusion processes Focusing on the “small scale”: network motifs –Networks as functioning circuits

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Regular networks

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Regular networks – fully connected Slides by Kraemer & Barabasi, Bonabeau (SciAm’03)

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Regular networks –Lattice

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Regular networks –Lattice: ring world

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Random networks

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Random networks (Erdos-Renyi, ‘60)

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Random Networks

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Small-world networks

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Small-world networks (Watts- Strogatz, ‘98)

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Small-world networks

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Scale-free networks

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A. fulgidus (archaea) Connectivity distributions for metabolic networks C. elegans (eukaryote) E. coli (bacterium) averaged over 43 organisms Jeong et al. Nature (2000)

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Protein-protein interaction networks Jeong et al. Nature 411, (2001) Wagner. RSL (2003)

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Preferential attachment model

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A simple model for generating “scale-free” networks 1.Evolution : networks expand continuously by the addition of new vertices, and 2.Preferential-attachment (rich get richer) : new vertices attach preferentially to sites that are already well connected. Barabasi and Albert. Science (1999) Barabasi & Bonabeau Sci. Am. May

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To incorporate the growing character of the network, starting with a small number (m 0 ) of vertices, at every time step we add a new vertex with m (< m 0 ) edges that link the new vertex to m different vertices already present in the system. Barabasi and Albert. Science (1999) Scale-free network model

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To incorporate preferential attachment, we assume that the probability P that a new vertex will be connected to vertex i depends on the connectivity k i of that vertex, so that (k i ) = k i / j k j. Barabasi and Albert. Science (1999) Scale-free network model

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This network evolves into a scale-invariant state with the probability that a vertex has k edges, following a power law with an exponent = 2.9 +/- 0.1 After t time steps, the model leads to a random network with t + m 0 vertices and mt edges. Barabasi and Albert. Science (1999) Scale-free network model

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Overview of part 1 Historical perspective –From reductionism to systems and networks Examples of complex networks Structural/topological metrics –Average path length –Degree distribution –Clustering Topological models –Regular, random, small-world, scale-free networks –An evolutionary model of network growth: Preferential attachment Implications of scale-free property in: –Robustness/fragility –Epidemics/diffusion processes Focusing on the “small scale”: network motifs –Networks as functioning circuits

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Robustness/fragility of scale-free networks

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Robustness/fragility

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Yeast protein-protein interaction networks Lethal Slow-growth Non-lethal Unknown Jeong et al. Nature 411, (2001) the phenotypic effect of removing the corresponding protein:

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Epidemics & other diffusion processes in scale-free networks

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Epidemics in complex networks

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Node dynamics and self-organization: Epidemics in complex networks

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If 3 we have absence of an epidemic threshold. If 4 an epidemic threshold appears, but it is approached with vanishing slope. If 4 the usual MF behavior is recovered. SF networks are equal to random graph. If 3 we have absence of an epidemic threshold. If 4 an epidemic threshold appears, but it is approached with vanishing slope. If 4 the usual MF behavior is recovered. SF networks are equal to random graph. Results can be generalized to generic scale-free connectivity distributions P(k)~ k - Pastor-Satorras & Vespignani (2001, 2002), Boguna, Pastor-Satorras, Vespignani (2003), Dezso & Barabasi (2001), Havlin et al. (2002), Barthélemy, Barrat, Pastor-Satorras, Vespignani (2004)

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Overview of part 1 Historical perspective –From reductionism to systems and networks Examples of complex networks Structural/topological metrics –Average path length –Degree distribution –Clustering Topological models –Regular, random, small-world, scale-free networks –An evolutionary model of network growth: Preferential attachment Implications of scale-free property in: –Robustness/fragility –Epidemics/diffusion processes Focusing on the “small scale”: network motifs –Networks as functioning circuits

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Reference Uri Alon, “An Introduction to Systems Biology: Design Principles of Biological Circuits”, Chapman & Hall, 2007

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Definition of motifs Network motifs are subgraphs that occur significantly more often in a real network than in the corresponding randomized network.

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Network motifs Original network Random version of original network

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Motifs in genetic network of yeast

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Motifs in genetic network of E. coli

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Examples of network motifs (3 nodes) Feed forward loop –Found in many transcriptional regulatory networks coherentincoherent

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Possible functional role of a coherent feed-forward loop Noise filtering: short pulses in input do not result in turning on Z

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Conservation of network motif constituents Homo SapiensMus musculus Drosophila melanogaster C. elegans Arabidopsis thaliana Saccharomyces cerevisiae Four nodes motif Orthologs

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